Consider the function f(x) = arctan(3x). Show that the graphy = f(x) and its tangent line y = g(x) at (0,0). Intermediate steps: 1) The line tangent to y = f(x) at x =0 is y = g(x) where g(x) = 2) Let H(x) = f(x) - g(x). The derivative of H (x) is H'(x) = = 0) intersect only at which is zero only when x = 3) Now assume that we have ₁ <0 where f(x₁) = g(x₁). Apply Rolle's theorem to H (x) on the interval [x1, 0]. Get a contradiction. 4) Now assume that we have 2 > 0 where f(2)= g(x2). Apply Rolle's theorem to H(x) on the interval [0, ₂]. Get a contradiction. 5) Conclude that the graph of f(x) and its tangent line intersect only at (0,0).

Consider the function f(x) = arctan(3x). Show that the graphy = f(x) and its tangent line y = g(x) at (0,0). Intermediate steps: 1) The line tangent to y = f(x) at x =0 is y = g(x) where g(x) = 2) Let H(x) = f(x) - g(x). The derivative of H (x) is H'(x) = = 0) intersect only at which is zero only when x = 3) Now assume that we have ₁ <0 where f(x₁) = g(x₁). Apply Rolle's theorem to H (x) on the interval [x1, 0]. Get a contradiction. 4) Now assume that we have 2 > 0 where f(2)= g(x2). Apply Rolle's theorem to H(x) on the interval [0, ₂]. Get a contradiction. 5) Conclude that the graph of f(x) and its tangent line intersect only at (0,0).

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.CR: Chapter 3 Review

Problem 8CR

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